Simplify the following expression and state the condition under which the simplification is valid: $a = \dfrac{t^2 - t - 56}{t^2 - 9t + 8}$
Explanation: First factor the expressions in the numerator and denominator. $ \dfrac{t^2 - t - 56}{t^2 - 9t + 8} = \dfrac{(t + 7)(t - 8)}{(t - 1)(t - 8)} $ Notice that the term $(t - 8)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(t - 8)$ gives: $a = \dfrac{t + 7}{t - 1}$ Since we divided by $(t - 8)$, $t \neq 8$. $a = \dfrac{t + 7}{t - 1}; \space t \neq 8$